Lecture 21
bench# A tibble: 4 × 6
expression min median `itr/sec` mem_alloc `gc/sec`
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl>
1 d[d$x > 0.5, ] 129µs 137µs 7203. 240.27KB 19.2
2 d[which(d$x > 0.5), ] 139µs 151µs 6577. 272.24KB 36.3
3 subset(d, x > 0.5) 170µs 192µs 5174. 289.27KB 26.1
4 filter(d, x > 0.5) 386µs 413µs 2375. 1.48MB 42.6
# A tibble: 4 × 6
expression min median `itr/sec` mem_alloc `gc/sec`
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl>
1 d[d$x > 0.5, ] 12ms 12.2ms 81.7 13.4MB 73.1
2 d[which(d$x > 0.5), ] 13.4ms 13.6ms 73.7 24.8MB 155.
3 subset(d, x > 0.5) 17.9ms 19.2ms 49.2 24.8MB 107.
4 filter(d, x > 0.5) 14.1ms 15.1ms 64.9 24.8MB 104.
bench - relative results# A tibble: 4 × 6
expression min median `itr/sec` mem_alloc `gc/sec`
<bch:expr> <dbl> <dbl> <dbl> <dbl> <dbl>
1 d[d$x > 0.5, ] 1 1 1.66 1 1
2 d[which(d$x > 0.5), ] 1.12 1.11 1.50 1.86 2.12
3 subset(d, x > 0.5) 1.50 1.57 1 1.86 1.46
4 filter(d, x > 0.5) 1.18 1.23 1.32 1.86 1.42
t.testImagine we have run 1000 experiments (rows), each of which collects data on 50 individuals (columns). The first 25 individuals in each experiment are assigned to group 1 and the rest to group 2.
The goal is to calculate the t-statistic for each experiment comparing group 1 to group 2.
# A tibble: 50 × 1,002
ind group exp1 exp2 exp3 exp4 exp5
<int> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
1 1 1 10.6 7.86 8.69 10.4 16.3
2 2 1 12.8 7.96 11.6 14.7 14.3
3 3 1 11.1 11.4 7.28 1.62 13.3
4 4 1 12.0 3.25 7.27 11.6 9.24
5 5 1 4.12 3.34 11.0 10.8 6.79
6 6 1 7.34 11.5 10.2 15.6 11.7
7 7 1 7.18 9.51 14.5 11.8 7.45
8 8 1 6.93 7.80 17.6 8.75 12.8
9 9 1 5.53 15.0 11.4 13.1 11.4
10 10 1 18.2 10.8 10.5 12.5 6.43
# ℹ 40 more rows
# ℹ 995 more variables: exp6 <dbl>, exp7 <dbl>,
# exp8 <dbl>, exp9 <dbl>, exp10 <dbl>,
# exp11 <dbl>, exp12 <dbl>, exp13 <dbl>,
# exp14 <dbl>, exp15 <dbl>, exp16 <dbl>,
# exp17 <dbl>, exp18 <dbl>, exp19 <dbl>,
# exp20 <dbl>, exp21 <dbl>, exp22 <dbl>, …
ttest_hand_calc = function(X) {
f = function(x, grp) {
t_stat = function(x) {
m = mean(x)
n = length(x)
var = sum((x - m) ^ 2) / (n - 1)
list(m = m, n = n, var = var)
}
g1 = t_stat(x[grp == 1])
g2 = t_stat(x[grp == 2])
se_total = sqrt(g1$var / g1$n + g2$var / g2$n)
(g1$m - g2$m) / se_total
}
apply(X[,-(1:2)], 2, f, X$group)
}
system.time(ttest_hand_calc(X)) user system elapsed
0.017 0.001 0.021
bench::mark(
ttest_formula(X, m),
ttest_for(X, m),
ttest_apply(X),
ttest_hand_calc(X),
check=FALSE
)Warning: Some expressions had a GC in every iteration; so filtering
is disabled.
# A tibble: 4 × 6
expression min median `itr/sec` mem_alloc `gc/sec`
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl>
1 ttest_formula(X, m) 197.85ms 208.46ms 4.87 8.24MB 24.3
2 ttest_for(X, m) 63.58ms 68.79ms 14.7 1.91MB 25.7
3 ttest_apply(X) 56.14ms 61.79ms 15.7 3.48MB 23.6
4 ttest_hand_calc(X) 8.68ms 9.69ms 84.9 3.44MB 25.7
parallelPart of the base packages in R
tools for the forking of R processes (some functions do not work on Windows)
Core functions:
detectCores
pvec
mclapply
mcparallel & mccollect
detectCoresSurprisingly, detects the number of cores of the current system.
Parallelization of a vectorized function call
user system elapsed
0.096 0.013 0.109
user system elapsed
0.166 0.159 0.258
user system elapsed
0.090 0.190 0.174
bench::system_timeParallelized version of lapply
Asynchronously evaluation of an R expression in a separate process
Checks mcparallel objects for completion
$`62243`
[1] 0.0005776226
Warning in selectChildren(jobs, timeout): cannot wait for child 62243
as it does not exist
NULL
Warning in selectChildren(jobs, timeout): cannot wait for child 62243
as it does not exist
NULL
Packages by Revolution Analytics that provides the foreach function which is a parallelizable for loop (and then some).
Core functions:
registerDoMC
foreach, %dopar%, %do%
registerDoMCPrimarily used to set the number of cores used by foreach, by default uses options("cores") or half the number of cores found by detectCores from the parallel package.
foreachA slightly more powerful version of base for loops (think for with an lapply flavor). Combined with %do% or %dopar% for single or multicore execution.
foreach - iteratorsforeach can iterate across more than one value, but it doesn’t do length coercion
foreach - combining resultsforeach - parallelizationSwapping out %do% for %dopar% will use the parallel backend.
user system elapsed
0.299 0.036 0.114
user system elapsed
0.312 0.052 0.082
user system elapsed
0.324 0.064 0.075
user system elapsed
0.000 0.000 3.008
user system elapsed
0.045 0.007 3.071
Bootstrapping is a resampling scheme where the original data is repeatedly reconstructed by taking a samples of size n (with replacement) from the original data, and using that to repeat an analysis procedure of interest. Below is an example of fitting a local regression (loess) to some synthetic data, we will construct a bootstrap prediction interval for this model.
Optimal use of parallelization / multiple cores is hard, there isn’t one best solution
Don’t underestimate the overhead cost
Experimentation is key
Measure it or it didn’t happen
Be aware of the trade off between developer time and run time
An awful lot of statistics is at its core linear algebra.
For example:
\[ \hat{\beta} = (X^T X)^{-1} X^Ty \]
Principle component analysis
Find \(T = XW\) where \(W\) is a matrix whose columns are the eigenvectors of \(X^TX\).
Often solved via SVD - Let \(X = U\Sigma W^T\) then \(T = U\Sigma\).
Not unique to Statistics, these are the type of problems that come up across all areas of numerical computing.
Numerical linear algebra \(\ne\) mathematical linear algebra
Efficiency and stability of numerical algorithms matter
Don’t reinvent the wheel - common core linear algebra tools (well defined API)
Low level algorithms for common linear algebra operations
Basic Linear Algebra Subprograms
Copying, scaling, multiplying vectors and matrices
Origins go back to 1979, written in Fortran
Linear Algebra Package
Higher level functionality building on BLAS.
Linear solvers, eigenvalues, and matrix decompositions
Origins go back to 1992, mostly Fortran (expanded on LINPACK, EISPACK)
Most default BLAS and LAPACK implementations (like R’s defaults) are somewhat dated
Written in Fortran and designed for a single cpu core
Certain (potentially non-optimal) hard coded defaults (e.g. block size).
Multithreaded alternatives:
ATLAS - Automatically Tuned Linear Algebra Software
OpenBLAS - fork of GotoBLAS from TACC at UTexas
Intel MKL - Math Kernel Library, part of Intel’s commercial compiler tools
cuBLAS / Magma - GPU libraries from Nvidia and UTK respectively
Accelerate / vecLib - Apple’s framework for GPU and multicore computing
| n | 1 core | 2 cores | 4 cores | 8 cores | 16 cores |
|---|---|---|---|---|---|
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 500 | 0.004 | 0.003 | 0.002 | 0.002 | 0.004 |
| 1000 | 0.028 | 0.016 | 0.010 | 0.007 | 0.009 |
| 2000 | 0.207 | 0.110 | 0.058 | 0.035 | 0.039 |
| 3000 | 0.679 | 0.352 | 0.183 | 0.103 | 0.081 |
| 4000 | 1.587 | 0.816 | 0.418 | 0.227 | 0.145 |
| 5000 | 3.104 | 1.583 | 0.807 | 0.453 | 0.266 |
Sta 523 - Fall 2024